Consider the two real-valued functions and .

MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

Consider the two real-valued functions

f(x)=|x|

and

g(x)=\operatorname{sgn}(x)

. (i) State the domain and range of each function. (ii) Determine all points of discontinuity, if any, of

g(x)

and justify your answer. (iii) Describe how the values of

f(x)

and

g(x)

change as

x

moves from negative to positive values across

x=0

. (iv) On the same pair of axes, sketch the graphs of

y=f(x)

and

y=g(x)

for

-4\le x\le 4

, clearly labelling the intercepts, the cusp or jump (if any), and indicating the scales on both axes.